1

Behavior of the anomalous correlation function in

uniform 2D Bose gas

Abdelâali Boudjemâa1

Department of Physics, Faculty of Sciences, Hassiba Benbouali University of Chlef

P.O. Box 151, 02000, Chlef, Algeria.

Abstract We investigate the behavior of the anomalous correlation function in two dimensional Bose gas. In the local case, we find that this quantity has a finite value in the limit of weak interactions at zero temperature. The effects of the anomalous density on some thermodynamic quantities are also considered. These effects can modify in particular the chemical potential, the ground sate energy, the depletion and the superfluid fraction. Our predictions are in good agreement with recent analytical and numerical calculations. We show also that the anomalous density presents a significant importance compared to the non-condensed one at zero temperature. The single-particle anomalous correlation function

is expressed in two dimensional homogenous Bose gases by using the density-phase fluctuation. We then confirm that the anomalous average accompanies in analogous manner the true condensate at zero temperature while it does not exist at finite temperature.

Key words: Homogenous Bose gas, two dimensions, Equation of State, Anomalous density. PACS: 05.30.Jp; 03.75.Hh; 11. 15. Tk

1 Corresponding author :e-mail: a.boudjemaa@univ-chlef.dz

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I. Introduction

The experimental progress of the ultracold gases in two dimensions (2d) [1-8]

has recently attracted great attention. The properties of these fluids are radically

different from those in three dimensions. The famous Mermin-Wagner-Hohenberg

theorem [9, 10] states that long-wavelength thermal fluctuations destroy long-range

order in a homogeneous one-dimensional Bose gas at all temperatures and in a

homogeneous two-dimensional Bose gas at any nonzero temperature, preventing

formation of condensate.

Since the earlier works of Schick [11] and Popov [12], several theoretical

studies of fluctuations, scattering properties and the appropriate thermodynamics have

been performed in [13-16]. In fact, in most of the previous references, the anomalous

density is neglected under the claim that it is a divergent and unmeasured quantity as

well as its contribution is very small compared to the other terms. Otherwise, the

importance of the anomalous density in three-dimensional Bose gas has been shown

in our recent theoretical results [17, 18] and also by several authors [19-24] using

different approaches. Theoretically, the anomalous average arises of the symmetry-

breaking assumption [17, 19, 24]. It quantifies the correlations of pairs of non-

condensate atoms with pairs of condensate atoms due to the Bogoliubov pair

promotion process in which two condensate atoms scatter each other out of the

condensate which is responsible for the well-known Bogoliubov particle-hole

structure of excitations in the system [24]. The anomalous density can also be

interpreted as a measure of the squeezing of the non-condensate field fluctuations

[25]. Certainly, the presence of this quantity adds new features to the well-known

problems and attracts our attention to the two-dimensional systems. A number of

questions arise naturally in this paper: Does the anomalous density exist even at finite

temperature in 2d Bose gas? How does its behavior compare with

Due to the complexity and the particularity of dilute 2d Bose gases, many analytical

investigations have been performed recently to find corrections beyond mean-field at

zero temperature. One should cite at this stage that Pricoupenko [26] employs the

pseudo potential with a Gaussian variational approach. Mora and Castin [27], on the

other hand, used their lattice model which is a sort of regularization scheme to treat

ultraviolet divergences. Cherny et al [28] used a reduced-density matrix of second

the normal density

at zero temperature? What are the effects of this quantity on the thermodynamics of

the system?

3

order and a variational procedure to derive results identical to those of Refs [26, 27]

for equation of state (EoS) and ground-state energy. The above analytical results have

been checked using Monte Carlo calculations to find numerical agreement with

beyond mean-field terms in 2d [29, 30]. Recently, Mora and Castin [31] have been

also extended their approach [27] one step beyond Bogoliubov theory which gives

good accuracy with the simulations of [30]. Another kind of extensions has been

developed recently by Sinner et al [32] which based on using the functional

renormalization group to study dynamical properties of the 2d Bose gas at 0=T . The

approach is free from infrared divergences and satisfies both the Hungeholtz-Pines

(HP) [33] relation and the Nepomnyashchy identity [34], which states that the

anomalous self-energy vanishes at zero frequency and momentum. The spectrum

energy thus satisfies a Bogoliubov-type expression with a renormalized sound

velocity. Although the above approaches provide good predictions for the

thermodynamic of 2d Bose gas in the universal regime, they are limited only at zero

temperature.

The present paper deals with extending our variational Time Dependent

Hartree-Fock-Bogoliubov (TDHFB) theory to the case of 2d Bose systems. The

theory was previously presented for 3d systems in [17, 18]. In fact, the

The paper is organized as follows: In Sec. II, we briefly review the derivation of the

TDHFB formalism, and give the different quantities which we study in 2d

homogeneous system. In Sec. III, we restrict ourselves to the behavior of the

anomalous density and its effects on the depletion, the chemical potential and the

ground-state energy. We therefore, compare our results with recent Monte-Carlo

simulations and analytic predictions. The validity of the HP theorem and

Nepomnyashchy identity are also discussed within the present formalism. In Sec. IV,

we extend our results at finite temperature where we calculate in particular the one-

body anomalous correlation function. In Sec. V, we apply our formalism to analyze

main

difference between our approach and the earlier variational HFB treatments is that in

our variational theory we do not minimize only the expectation values of a single

operator like the free energy in the standard HFB approximation. Conversely, our

variational theory is based on the minimization of an action also with a Gaussian

variational ansatz. The action to minimize involves two types of variational objects:

one related to the observables of interest and the other akin to a density matrix [35,

36].

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the behavior of the superfluid fraction. We then emphasize the importance of the

anomalous density for the occurrence of superfluid transition and sound velocity. Our

conclusion and perspectives are drawn in Sec. VI.

II. Formalism

Our starting point is the TDHFB equations which describe the dynamics of d-

dimensional interacting trapped Bose systems. For a short-range interaction potential

and sufficiently dilute gas, the TDHFB equations read.

( ) ( ) ( )[ ] ( ) ( ),*~)(~ rrmgrrngrgnhri ddcsp Φ+Φ++=Φ 2

(1.a)

+ℑ−ℑ= ρρρdtdi , (1.b)

where µ−+∆−= )(2 ext

2

rVm

hsp is the single-particle Hamiltonian, )(ext rV

is the

external trapping potential, µ is the chemical potential and dg is the interaction

parameter in d-dimensions.

Here we have defined the 2×2 matrices

( ) ( ) ( )( ) ( )

′−∆−

′′∆′=′ℑ ∗∗ rrhrr

rrrrhrr

,,,,

.

and

( ) ( ) ( )( ) ( ) ( )

′δ+′′

′′=′ρ ∗∗ rrrrnrrm

rrmrrnrr

,,~,~,~,~

. ,

where ( ) ( ) ( ) ( ) ( )[ ]'',~, rrrrngrhrrh dsp

Φ′Φ+′+=′ ∗2 , ( ) ( ) ( ) ( )[ ]rrrrmgrr d

ΦΦ+=∆ ,~,

and

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )''),'(~)',(~'')',(~)',(~

rrrrrrmrrmrrrrrrnrrn

ΦΦ−>ΨΨ=<≡ΦΦ−>ΨΨ=<≡ ∗+∗

, (2)

are respectively the normal and anomalous single-particle correlation functions. In the

local case they play the role of the non-condensed and anomalous densities.Moreover,

our formalism provides a direct link between these two later quantities as

( ) ( ) ( ) ( ) ( )[ ]∫ ′′′′′−′′′′′′′=′ rrrrrrrrrdrrI

,,,,, 21122211 ρρρρ . (3)

Notice that Eq.(3) is often known as the Heisenberg invariant, it is a direct

consequence of the conservation of the von-Neumann entropy DD ln Tr −=S . For

pure state Eq.(3) takes the form ( ) ( )rrrrI d ′−=′ δ, [37].

5

Among the advantages of the TDHFB equations is that the three densities are coupled

in a consistent and closed way. Second, they should in principle yield the general

time, space and temperature dependence of the various densities. Furthermore, they

satisfy the energy and number conserving laws. In addition, the most important

feature of the TDHFB equations is that they are valid for any Hamiltonian H and for

any density matrix operator. Interestingly, our TDHFB equations can be extended to

provide self-consistent equations of motion for the triplet correlation function by

using the post Gaussian ansatz.

In the uniform case ( ( ) 0ext =rV ) and for a thermal distribution at equilibrium,

by working in the momentum space,

( )( )

( ) ( )kekdrr ijrrki

d

d

ij ρπ

ρ

′−∫=′ ., 2

, (4)

where ( )kijρ is the Fourier transform of ( )rrij ′,ρ . We can then easily rewrite Eq. (3)

as

( ) ( )TmnnI

kkkkk 2/sinh4

1~1~~2

2

ε=−+= , (5)

with kε is the Bogoliubov energy spectrum given below.

The physical meaning of Eq.(5) is that it allows us to calculate in a very useful way

the dissipated heat for d-dimensional

( )d

d

kkdkdIE

nQ

π= ∫ 2

1

Bose gas as

, (6)

where mkEk 222 /= is the energy of a free particle.

Furthermore, a )1~(~~ 2 += kkk nnmt zero temperature, Eq. (5) reduces to , which

constitutes an explicit relationship between the normal and the anomalous densities

at zero temperature and indicates that these two quantities are of the same order of

magnitude at low temperatures which leads to the fact that neglecting m~ while

maintaining n~ is a quite unsafe approximation.

The excitation energy kε is determined in our formalism via the random-phase

approximation (RPA) [36] which can be found by expanding all quantities around

their equilibrium solution. The RPA appears as a direct application of the general

Balian-Vénéroni formalism to the Lie algebra of single boson operators [36, 37].

Thus we write

6

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )tkmkmtkm

tknkntkntkktk

,~~,~,~~,~

,,

δδδ

+=+=

Φ+Φ=Φ , (7)

Then, we have written these quantities on a diagonal basis (RPA matrix) which

derived from the set (3), and kept only the first-order terms. After a long, but

straightforward calculation, we arrive at the gapless expression of the Bogoliubov

spectrum [18, 38]

( ) ,- E 212

211k Σ−Σ+µ=εk (8)

where nd11 g2=Σ and ( )mnc~gd12 +=Σ are respectively the first order normal and

anomalous self-energies with nnn c~+= is the total density.

A detailed derivation of the Bogoliubov spectrum with the RPA method will be given

elsewhere.

Note that Eq.(8) can be also obtained using the Green’s functions (see e.g.[38]). It

provides a useful finite-temperature version of the healing length and the sound

velocity sc as

sc

dc mcnmgmnm /~

1// 12 =

+=Σ=ξ , (9)

In order to get explicit formulas for the non-condensed and the anomalous averages in

d-dimensions we may use Eq. (5). A simple calculation yields

( )( )

∫

−

++=

−1

~

221~

1 kk

cdkd

d

ImngE

L

kdnεπ

, (10.a)

( )( )

∫

+−=

− kk

cdd

d

Imng

L

kdmεπ

~

221~

1, (10.b)

It is worth noting that Eqs. (1.a) and (10) together form the generalized HFB

equations. This shows that, in the static case, our formalism recovers easily the full

HFB equations at both finite and zero temperatures.

III. Anomalous density at zero temperature

Let us now discuss the behavior of the normal and anomalous densities in

homogeneous 2d Bose gas at both zero and finite temperatures. From this point we

consider the regime in weakly repulsive interaction at zero temperature

1=kI

where . In two-dimensional Bose gas, the interaction parameter ( 2ggd = )

depends logarithmically on the chemical potential as

7

( ) /2ln

1422

2

2

µ

π=

ammg

, (11)

where a is the two-dimensional scattering length among the particles and 2g is the

two-body T-matrix (see e.g. [15,26,27]).

The calculation of the integral in Eq. (10.a) leads us to the following expression of the

depletion

2 41

ξπ nnn

=~

.

This equation is in good agreement with that obtained in[26].

(12)

The integral in Eq.(10.b) has an ultraviolet divergence in both two and three

dimensions. This divergence is well-known and arises due to the use of the contact

potential. To regulate the ultraviolet divergences, we may use the dimensional

regularization [39-40]. In such a technique one calculates the loop integrals in

η22 −=d dimensions for values of η where the integrals converge. One then

analytically continues back to 2=d dimensions. With dimensional regularization, an

arbitrary renormalization scale M is introduced. This scale can be identified with the

simple momentum cutoff. An advantage of dimensional regularization is that in two

dimensions systems it automatically sets power divergences to zero, while logarithmic

divergences show up as poles in η [40]. Using this technique one gets for the

anomalous density

10

2

0 4 ,~ JmT

Λ−=Λ

= , (13)

where Λ is the regularized-part which is related to the size of particles and interactions

as ξ=Λ /2 . This parameter is similar to that used in [26, 32, 40].

And ( )

+−= η

ηπOLJ 1

41

10 , with ( )22 4/ln ML Λ= .

Thus the convergent part of the anomalous density provides

( )222

0 4/ln16

~ MmT Λπ

Λ=Λ

= . (14)

A useful remark at this level, the non-condensed density of Eq.(12) can be rewritten

also in terms of Λ as πΛ= 16/~ 2n .

At 0=T , the condensed density has a significant value and hence constitutes the

dominant quantity in the system, while both the non-condensed and the anomalous

8

densities vanish for 0→Λ which ensures that µ=Σ =Λ 2011 and µ=Σ =Λ 0

12 in good

accordance with the Hugenholtz-Pines theorem µ=Σ−Σ =Λ=Λ 012

011 [33]. On the other

hand, once we find 0012 ≠Σ =Λ , this means that the actual version of our extended

variational TDHFB including a dimensional regularization does not satisfy the

Nepomnyashchy identity [32] as it should be for any limited approximation order (see

for example Griffin and Shi [38], Yukalov [19], Pricoupenko [26] and Andersen

[40]). Indeed, in a Bose-condensed system, the anomalous self-energy must be

nonzero in order to define a meaningful nonzero sound velocity and healing length.

In addition, a zero sound velocity leads evidently to an unstable system (see Eq.(9)).

Indeed, our approach can be an effective way to verify the Nepomnyashchy identity,

but on the condition that we sum over all terms of perturbation theory for the self-

energy with renormalization of the sound velocity to ensure the stabilization of the

system as it has been demonstrated in [32].

Conversely, at finite-temperature three-dimensional Bose gas, the chemical potential

satisfies the generalized version of HP theorem given by Hohenberg and Martin

[41] µ=Σ−Σ ΛΛ1211 . In such a situation, when cTT ≥ , both the condensate and the

anomalous densities vanish [17-19, 24, 25] whatever the value of Λ , which implies

directly that 012 =ΣΛ and µ≈=ΣΛ 2~211 ng . Consequently, a vanishing anomalous self-

energy is further guaranteed at high temperature and momentum in 3d systems.

Physically, this result is reasonable because the gas becomes completely thermalized

and therefore there is neither superfluid nor acoustic waves when the temperature

reaches its critical value.

On the other side, the dimensional regularization gives an asymptotically exact result

at weak interactions ( 02 →g ). The extrapolation to finite interactions requires that the

limiting condition 1/~ <<cnm be verified. In real systems, however, the interactions

have a finite range a and so a/1 provides a natural ultraviolet cutoff M [40]. Hence,

using this technique presupposes that value of the healing length in Eqs.(12) and (13)

takes the form 2gmnc/=ξ . In the case where nnc ≈ , we recover easily the well-

known result of Schick [11] for the depletion

µπµ

== 2

20

20

0 ln4

~

ammmT

, while Eq.(14) has no analog in the

literature. Under these conditions, the anomalous average turns out to be given as

, (15)

9

where ng d20 =µ .

FIG.1 shows that the non-condensed and the anomalous densities, as function of the

dimensionless parameter 220 /amx µ= , are competitive contributions at zero

temperature. For small values of x , we observe that the non-condensed density is

greater than the anomalous one while this later becomes the dominant quantity for the

whole range of x ( 10>x ). n~ and m~ are comparable only for 10≈x . Therefore, we

deduce that omitting the anomalous density, while keeping the normal one is

physically and mathematically inappropriate. It is noticed that this behavior holds also

in three dimensional Bose gas [17-25].

0 5 10 15 20 25 300

10

20

30

40

x

Norm

al a

nd a

nom

alou

s de

nsitie

s (u

nits

of a

2 )

FIG 1. Non-condensed (dashed line) and anomalous (solid line) densities as function of x

It is important now to discuss how the anomalous density can modify the

chemical potential and hence

( )mngd~~ +=δµ

the other thermodynamic quantities of dilute Bose gas.

The first order quantum corrections to the chemical potential are given

by [42]. Therefore, using the results obtained in Eqs.(12) and (15)

with the assumption cnn ≈ at 0=T . One obtains for the chemical potential

µπ

µ+µ=µ 2

20

20

0 ln4

1

eamn

m , (16)

The ground-state energy is obtained through

µπ

+µ

=µ= ∫ 2

20

220

0

ln8

12

/

eammgdnNEn

, (17)

10

The leading term in Eq.(16) was first obtained by Schick [11] while the second

represents our correction to the chemical potential. Clearly this correction is

universal, depending only on the interactions and scattering length. It is worth

mentioning that the additional logarithm term in Eq.(16) is analogous to that found

recently in [31]. Moreover, what is interesting in

Before plotting figures (2) and (3), we use the dimensionless relation

Eq. (16) is that if we invert it and

take the limit of vanishing density, we thereby recover the well-known Popov's EoS

[12].

−=

γ

2ln

2122 xexna where

γ is the Euler’s constant [26, 31].

0.00 0.05 0.10 0.15 0.20

51051104

51040.001

0.0050.010

x

na2

FIG. 2. Equation of State of 2D homogeneous Bose gas. Solid line: our extended variational approach. Dashed line: EoS predicted in [26, 27]. Dotted line: first correction beyond Bogoliubov theory [31].

11

FIG. 3. Ground state energy of a two dimensional Bose gas, as a function of the gas density, in units of the mean field prediction EMF

. Brown solid line: our calculations. Dark solid line: energy obtained from the beyond Bogoliubov [31]. Dashed line: analytical prediction of [28]. Dotted line: analytical calculations of [26]. Plotting symbols with error bars: numerical results of [29], for interactions given by hard disks (crosses) and by soft disks (circles); numerical results of [30], for dipolar interactions (diamonds).

One can see from FIG.2 that for the value of the gas parameter around 5×10-5

2na

there is

a difference of 10% between the equations of state while for larger than 5×10-3

FIG.3 shows that our expression of the ground sate energy (Eq.(17)) gives also an

estimate compatibility within error bars of Monte-Carlo simulations [29, 30] and

analytic results of [26-28, 31]. Furthermore, it is clearly seen from FIGs.2 and 3 that

there is an upward shift of our curve relative to that of ref [31], which is indeed due to

a prefactor which appears in the expansion of the later reference.

all

EoS are practically identical.

Another important feature revealed in FIG.3 is that there is no difference between

dipolar and short-range interaction for densities lower than 10-10

[30].

IV. Anomalous density at finite temperature

Now we turn to analyze the finite temperature case. As we mentioned in the

introduction, the finite temperature uniform 2d Bose gas is characterized by the

absence of a true Bose-Einstein condensate and long-range order [9, 10]. So the

physics of 2d Bose gas at finite-T can be understood in the context of the density-

phase representation. Accordingly, φien=Ψ

the single-particle anomalous correlation function

is found by using the field operator in the form: . Following the

hydrodynamic approach described in [14, 15, 43] with the assumptions 1/~ <<cnm and

12

nnc ≈ for 0→T .

( ) ( )

−= ∫ 2

.cos2/coth21exp0,~ 2 rkT

Ekd

nnrm k

k

k

εε

Then on the basis of Eq.(2), we obtain for the single-particle

anomalous correlation function

At low temperatures

, (18)

( µ<<T ) the main contribution to the integral of Eq.(18) comes

from the region of small momentum then the single-particle anomalous correlation

function undergoes a slow law decay at large distances:

( )dTT

rnrm

2/~

=

ξ , (19)

where

mnTd

22 π= is the temperature of quantum degeneracy.

We can infer from these results that the anomalous average does not exist at finite

temperature. This is ( )rm~strictly confirmed by Eq.(19) where one finds that vanishes

for ∞→r . Similarly to the situation in 2d at finite temperature regarding the normal

correlator ( ) ( ) ( ) ( )' ')',(~ rrrrrrn

ΦΦ−>ΨΨ=< ∗+ , which tends to zero as ∞→r ,

confirming that there is no true condensate, but one identifies instead the existence of

a quasicondensate. However, this result also implies that there is no symmetry

breaking, and consequently the anomalous average should not exist at any nonzero

temperature. The butter of this result is that the anomalous density accompanies in a

manner analogous the true condensate in a system of 2d homogeneous Bose gas.

V. Superfluid fraction

Usually, Bose-Einstein condensation is accompanied by superfluidity. However, in a

two-dimensional system at finite temperature, there is no BEC, but there still exists

superfluidity. The relation between them depends on the Bogoliubov-type

nature of the spectrum Eq.(7) [13]. Also, what is important is that our formalism

provides a useful relation between the superfluid fraction and the dissipated heat

which is equivalent to that obtained in Refs [13, 19].

( ) ( ) TQ

eekkd

mnTnnf d

T

Ts

sk

k2

2/

/2

2

22

1122

1 =

ε

ε

−=−π

−== ∫ , (20)

where sn is the superfluid density.

13

It is very important to mention that sf the superfluid fraction will be a divergent

quantity and thus the superfluid transition does not occur when the anomalous average

is omitted in Eq.(20).

At low temperature and weak interaction, we get

( ) 3422

331 Tmnc

fs

sπς

−= , (21)

where ( )3ς is a Riemann zeta function and the sound velocity turns out to be given

( )

+−=

nm

nncc ss

~~10 , (22)

with ( ) mcs /00 µ= is the zero order sound velocity.

Upon neglecting the normal and the anomalous fractions we recover straightforwardly

the superfluid fraction obtained earlier by Popov [12] and by Fisher and Hohenberg

[13].

VI. Conclusion

We have studied in this paper the behavior of the anomalous density in two-

dimensional homogeneous Bose gases. We find that this quantity has a finite value in

the limit of weak interactions. We have discussed also the effects of the anomalous

average on some thermodynamic quantities. As an example, we have given formulas

for the chemical potential, ground-state energy, the depletion and superfluid fraction.

The later does not occur if the anomalous density is neglected. In the ultra-dilute limit,

the known results are reproduced. This feature makes our predictions in accordance

with Monte-Carlo simulations and analytical calculations. Also, we have shown that

our approach satisfies the HP theorem at zero temperature while it does not verify the

Nepomnyashchy identity as it should be for any limited approximation. Moreover, the

importance of the anomalous density compared to the normal one at low temperature

has been also highlighted. In addition, by using the density phase fluctuation we

found that the single-particle anomalous correlation function undergoes a slow law

decay at large distances such a result implies that the anomalous average does not

exist at finite temperature.

Finally, an interesting question to ask is whether some quantity can exist in this

system which accompanies the quasicondensate density in a manner analogous to that

in which the anomalous density accompanies a true condensate?

14

The goal of our next work is to use our approach to answer this important question.

On the other hand we will try to extract something useful about superfluidity in 2d

Bose system. The idea is to relate our predictions with appropriate numerical

simulations for some realistic experiments in traps to study for example the vortex

stability without rotating fluid.

Acknowledgments

We acknowledge Ludovic Pricoupenko, Gora Shlyapnikov, Jean Dalibard and Usama Al-Khawaja for many useful comments about this work. We are grateful to J. Andersen and V. Yukalov for helpful discussions. We are indebted to Yvan Castin for giving us the numerical data. References [1] A. Görlitz et al., Phys. Rev. Lett. 87, 130402 (2001). [2] S. Burger et al., Europhys. Lett. 57, 1 (2002). [3] M. Hammes, D. Rychtarik, B. Engeser, H.C. Nagerl, R. Grimm, Phys. Rev. Lett. 90, 173001 (2003). [4] Y. Colombe et al., J. Opt. B 5, S155 (2003). [5] Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, and J. Dalibard, Nature 441, 1118 (2006). [6] P. Cladé, C. Ryu, A. Ramanathan, K. Helmerson, and W. D.Phillips, Phys. Rev. Lett. 102, 170401 (2009). [7] C.-L. Hung, X. Zhang, N. Gemelke, and C. Chin, Nature 470, 236 (2011). [8] T. Yefsah, R. Desbuquois, L. Chomaz, K. J. Gunter, and J. Dalibard, Phys. Rev. Lett. 107, 130401 (2011) . [9] N. D. Mermin, and H. Wagner, Phys. Rev. Lett. 22, 1133 (1966). [10] P. C. Hohenberg, Phys. Rev. 158, 383 (1967). [11] M. Schick, Phys. Rev. A3, 1067 (1971). [12] V. N. Popov, Theor. Math. Phys. 11, 565 (1972); Functional Integrals in Quantum Field Theory and Statistical Physics, (Reidel, Dordrecht, 1983), Chap. 6. [13] D. S. Fisher and P. C. Hohenberg , Phys. Rev. B37, 4936 (1988). [14] D. S. Petrov, G.V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 85, 3745 (2000). [15] U. Al Khawaja, J.O. Andersen, N.P. Proukakis, H.T.C Stoof, Phys. Rev. A 66,.

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